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Disproof of the Odd Hadwiger Conjecture

Marcus Kühn, Lisa Sauermann, Raphael Steiner, Yuval Wigderson

TL;DR

This work disproves the odd Hadwiger conjecture by constructing, via a careful two-layer overlay of triangle-free random graphs and a complement operation, an n-vertex graph G with α(G) ≤ 2 and χ(G) ≥ (3/2 − δ)n that does not contain K_t as an odd minor for large t. The authors develop a probabilistic framework with balanced random graphs and ε-respectful pairings, and prove that no large odd connected pairing can occur w.h.p., which blocks large odd minors. Central to the argument are two-layer blowup constructions, precise control of triangles, and a cascade of concentration and coupling lemmas that together bound the existence of odd minors. The results not only refute the conjecture in a strong form but also clarify the tightness of the approach, illustrating a fundamental barrier at the 3/2 factor for graphs with independence number 2.

Abstract

We prove that there exist graphs which do not contain $K_t$ as an odd minor and whose chromatic number is at least $(\frac 32-o(1))t$. This disproves, in a strong form, the odd Hadwiger conjecture of Gerards and Seymour from 1993.

Disproof of the Odd Hadwiger Conjecture

TL;DR

This work disproves the odd Hadwiger conjecture by constructing, via a careful two-layer overlay of triangle-free random graphs and a complement operation, an n-vertex graph G with α(G) ≤ 2 and χ(G) ≥ (3/2 − δ)n that does not contain K_t as an odd minor for large t. The authors develop a probabilistic framework with balanced random graphs and ε-respectful pairings, and prove that no large odd connected pairing can occur w.h.p., which blocks large odd minors. Central to the argument are two-layer blowup constructions, precise control of triangles, and a cascade of concentration and coupling lemmas that together bound the existence of odd minors. The results not only refute the conjecture in a strong form but also clarify the tightness of the approach, illustrating a fundamental barrier at the 3/2 factor for graphs with independence number 2.

Abstract

We prove that there exist graphs which do not contain as an odd minor and whose chromatic number is at least . This disproves, in a strong form, the odd Hadwiger conjecture of Gerards and Seymour from 1993.
Paper Structure (15 sections, 14 theorems, 40 equations)

This paper contains 15 sections, 14 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Heuristics: pseudorandom triangle-free graphs
  3. The construction
  4. Probabilistic lemmas
  5. Proof of the main theorems
  6. Combinatorial lemmas
  7. Proof of Theorem \ref{['thm:odd pairing']}
  8. Proof of Theorems \ref{['thm:main']} and \ref{['thm:main alpha=2']}
  9. Properties of the random functions
  10. Properties of the random graphs
  11. A preparatory lemma
  12. Proof of Lemma \ref{['lemma: one connection survives']}
  13. Proof of Lemmas \ref{['lem:easy chernoff']} and \ref{['lem:easy chernoff triangles']}
  14. Concluding remarks
  15. Optimality of the factor 3/2

Key Result

Theorem 1.3

For any fixed $\delta>0$ and every integer $t$ which is sufficiently large with respect to $\delta$, there is a graph $G$ with $\chi(G) \geqslant (\frac{3}{2} -\delta)t$ which does not have $K_t$ as an odd minor. In particular, for every sufficiently large $t$, there is a graph with $\chi(G)\geqslan

Theorems & Definitions (27)

  • Conjecture 1.1: Hadwiger's conjecture
  • Conjecture 1.2: Odd Hadwiger conjecture
  • Theorem 1.3
  • Theorem 1.4
  • Proposition 1.5
  • Theorem 3.1
  • Lemma 4.1
  • Lemma 4.2
  • Lemma 4.3
  • Lemma 4.4
  • ...and 17 more